a) \( \frac{20^{3} \times 25^{2} \times 4}{10^{9}} \)
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First, let's simplify the expression step by step. We know that \(20 = 2^2 \times 5\), \(25 = 5^2\), and \(4 = 2^2\), so we can express everything in terms of prime factors: - \(20^{3} = (2^2 \times 5)^{3} = 2^{6} \times 5^{3}\) - \(25^{2} = (5^{2})^{2} = 5^{4}\) - \(10^{9} = (2 \times 5)^{9} = 2^{9} \times 5^{9}\) Now, substituting these in the original expression: \[ \frac{2^{6} \times 5^{3} \times 5^{4} \times 2^{2}}{2^{9} \times 5^{9}} \] Combining the powers of 2 and 5: \[ = \frac{2^{8} \times 5^{7}}{2^{9} \times 5^{9}} \] Now, subtract the powers in the fraction: \[ = 2^{8-9} \times 5^{7-9} = 2^{-1} \times 5^{-2} \] This can be rewritten as: \[ = \frac{1}{2} \times \frac{1}{5^{2}} = \frac{1}{2 \times 25} = \frac{1}{50} \] Thus, the simplified value of the expression \( \frac{20^{3} \times 25^{2} \times 4}{10^{9}} \) is \( \frac{1}{50} \).
